Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 16 min ago
PreprintCC BY-NC-ND 4.0arXiv

Resolvent comparison for Hermitian block matrices and Kelmans transformations

  • Xianhao Meng
  • Zhijun Xu
  • Zhiyuan Zhang
Open at sourcePDFDOICite

Abstract

A Kelmans transformation increases or preserves the largest eigenvalue of a symmetric nonnegative matrix. We determine when this comparison holds for Hermitian block extensions with a fixed coupling and an arbitrary common complementary block. An orthogonal change of basis and Schur complementation give an exact formula for the compressed resolvent difference as a difference of two rank-one matrices. This formula characterizes admissible coupling subspaces by proportionality of projected resolvent vectors. Polynomial coefficient kernels describe all such subspaces, and cyclic subspaces give dimension bounds and a maximum-dimension result for a two-block family. We also prove a general block comparison theorem and give monotonicity and equality criteria for nonnegative rank-one couplings. Applications concern graph joins, complex attachments, and Hermitian adjacency matrices.

Related

  1. Quantitative Resolvent Calculus for Sample Covariance Matrices
    Thomas Stark, Nina Dörnemann
    arXivPreprintarXiv licenceFull textProbabilityStatisticsarXiv
  2. Linear transformations of tropical matrices preserving individual values of the cyclicity index
    Alexander Guterman, Alexander Vlasov, Svetlana Zhilina
    arXivPreprintarXiv licenceFull textAlgebra & number theoryCombinatorics & discrete mathematicsarXiv
  3. Diagonalization of skew-Hermitian dual quaternion matrices and ϕ-Hermitian dual quaternion matrices
    Cailing Yao, Bingzhe Hou
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  4. Graph transformations and eigenvectors of the Laplacian, signless Laplacian and Adjacency matrices
    Jean-Guy Caputo, Leonardo de Lima, Arnaud Knippel
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsMathematical physicsarXiv
  5. Valuated delta matroids and principal minors of Hermitian matrices
    Nathan Cheung, Tracy Chin, Gaku Liu +1
    Algebraic CombinatoricsPeer-reviewedCC BY 4.0Full textCombinatorics & discrete mathematicsCrossref

Cite

BibTeXRIS
BibTeX
@misc{meng2026resolvent,
  title = {{Resolvent comparison for Hermitian block matrices and Kelmans transformations}},
  author = {Xianhao Meng and Zhijun Xu and Zhiyuan Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.04688},
  eprint = {2611.04688},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.04688},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Resolvent comparison for Hermitian block matrices and Kelmans transformations
AU  - Xianhao Meng
AU  - Zhijun Xu
AU  - Zhiyuan Zhang
PY  - 2026
DA  - 2026/11/03
PB  - arXiv
DO  - 10.48550/arxiv.2611.04688
UR  - https://arxiv.org/abs/2611.04688
AB  - A Kelmans transformation increases or preserves the largest eigenvalue of a symmetric nonnegative matrix. We determine when this comparison holds for Hermitian block extensions with a fixed coupling and an arbitrary common complementary block. An orthogonal change of basis and Schur complementation give an exact formula for the compressed resolvent difference as a difference of two rank-one matrices. This formula characterizes admissible coupling subspaces by proportionality of projected resolvent vectors. Polynomial coefficient kernels describe all such subspaces, and cyclic subspaces give dimension bounds and a maximum-dimension result for a two-block family. We also prove a general block comparison theorem and give monotonicity and equality criteria for nonnegative rank-one couplings. Applications concern graph joins, complex attachments, and Hermitian adjacency matrices.
ER  -